Splitting field of a central simple algebra

Remarks

Example: $\C$ splits $\mathbb{H}$ over $\R$. Indeed, $\mathbb{H} \otimes_\R \C$ is central simple of dimension $4$ over $\C$, and the only such algebra is $M_2(\C)$, since $\C$ is algebraically closed.

Non-example: $\R$ itself does not split $\mathbb{H}$, since $\mathbb{H} \otimes_\R \R \cong \mathbb{H}$ is a division algebra and $M_2(\R)$ has zero divisors.

$A$ is split over $K$ itself exactly when $A \cong M_n(K)$. An algebraic closure of $K$ splits every central simple algebra over $K$, and so does a suitable finite Galois extension of $K$. Every maximal subfield of a central division algebra $D$ is a splitting field of $D$.